qid int64 1 4.65M | question large_stringlengths 27 36.3k | author large_stringlengths 3 36 | author_id int64 -1 1.16M | answer large_stringlengths 18 63k |
|---|---|---|---|---|
2,934,529 | <p>I am trying to compute
<span class="math-container">$$\large \lim_{x\to \frac32^-}\frac{x^2+1}{3x-2x^2}$$</span> </p>
<p>I know the numerator will equal one fine and how it becomes <span class="math-container">$x^2+1 * 1/(3x-2x^2)$</span> because <span class="math-container">$a/b = a * 1/b$</span>, but then I'm not... | Ross Millikan | 1,827 | <p>The numerator will not equal <span class="math-container">$1$</span>, it will approach <span class="math-container">$1+(\frac 32)^2=\frac {13}{4}$</span>. The denominator will approach <span class="math-container">$0$</span> but always be positive because <span class="math-container">$x \lt \frac 32$</span>. The r... |
45,429 | <p>I have two problems: </p>
<p><strong>1.-</strong> Let $X$ be a compact Hausdorff space, then $X$ has a basis with cardinality less than or equal to $|X|$.</p>
<p><strong>2.-</strong> Let $X$ be a Hausdorff space and $D$ a dense subset in $X$, then $|X|\leq|P(P(D))|$, where $P(D)$ is the power set of $D$. </p>
<p... | Hunter Spink | 12,127 | <p>Let $A_q$ be the element in $2^{P(D)}$ which associates with each powerset $D$ a $1$ if the closure contains $q$, and $0$ otherwise. Then I claim that $A_q$ is unique for each point $q$ in the Hausdorff space, which proves the claim since
$$|2^{P(D)}|=|P(P(D))|$$</p>
<p>If $A_p=A_q$, then consider disjoint neighbor... |
847,719 | <p>How is the derivative of $(5x-2)^3$ equal to $15(5x-2)^2$ and not $3(5x-2)^2$. According to $\frac{df}{dx} = nx^{n-1}$, it has to be $3(5x-2)^2$ right. Please explain.</p>
| Community | -1 | <p>No, $$\frac{d}{dx} x^n = n x^{n - 1}$$ only applies to <em>pure</em> powers of $x$. $5x - 2$ is not a pure power of $x$, so this does not apply. The chain rule is necessary here; the derivative of the inner function ($5x - 2$) is $5$ here, accounting for the additional factor.</p>
|
844,403 | <p><img src="https://i.stack.imgur.com/iYn8g.png" alt=" Circumcircles, Incircles, Medians, and Altitudes" /></p>
<p>I tried to use the angle property by which AD=4 and DB=5,but since F is not given as mid point I don't know how to proceed to find length of DG.I think AED as 90 degree is important but I am unable to fig... | samarth srivastava | 144,402 | <p>CONSTRUCTION : extend $AE$ to $BC$ where it intersects $BC$ at $X$.</p>
<p>consider $\triangle AEC$ and $\triangle XCE$.</p>
<p>$\triangle AEC$ and $\triangle XCE$ are congruent by $ASA$.</p>
<p>therefore by CSCTC, $AE = EX$</p>
<p>therefore $E$ is the mid-point of $AX$.</p>
<p>given: $EF$ and $EG$ are parallel... |
2,122,472 | <p>If $x^2+x\cos(A + B) + 1$ is a factor of the expression,
$$2x^4 + 4x^3\sin A\sin B -x^2(\cos 2A +\cos 2B) + 4x\cos A\cos B -2$$
Then we have to find the other factor (s).</p>
<p>I am not getting start</p>
<p>Can anybody provide me a hint . </p>
| G Cab | 317,234 | <p>Then the other factor shall be a quadratic polynomial
that can be generally formulated as
$$
C\,x^{\,2} + D\,x + E
$$
So you must have
$$
\begin{gathered}
\left( {C\,x^{\,2} + D\, x + E} \right)\left( {\,x^{\,2} + \,\cos (A + B)\,x + 1} \right) = \hfill \\
= 2x^{\,4} + \left( {4\sin A\sin B} \right)\,x^{\,... |
1,302,383 | <p>Can anyone help me find the solution to this integral:</p>
<p>$$\int\limits{(t-4)(t-2)^{4/5}}dt?$$</p>
<p>I think I need to expand the integrand but I do not know how. Thanks a lot!</p>
| Mythomorphic | 152,277 | <p><strong>Hint:</strong></p>
<p>$$\int(t-4)(t-2)^{4/5}dt=\int(t-2)^{9/5}-2(t-2)^{4/5}d(t-2)$$</p>
|
493,862 | <p>How do I show that if $G$ is a cyclic group, say $G=\langle g \rangle$, then there exists a surjective homomorphism from the set of integers to $G$?</p>
<p>Do I start by listing the elements of $G=\langle g \rangle$ as $\lbrace 1, g, g^2, ..., g^{n-1}\rbrace$?</p>
| oxeimon | 36,152 | <p>Any finite cyclic group is isomorphic to $\mathbb{Z}/N\mathbb{Z}$, where $N$ is the order of the cyclic group. Any infinite cyclic group is isomorphic to $\mathbb{Z}$. In the first case, the surjective homomorphism is just the quotient map $\mathbb{Z}\rightarrow\mathbb{Z}/N\mathbb{Z}$. In the second case, the surjec... |
387,542 | <p>e.g. The function $e^x$ reflected through $y=x$ is $\ln x$. Is this always true OR just in some cases?</p>
| user152370 | 152,370 | <p>Suppose that $D(a,b)$ is a point on the graph of a one-to-one function defined by $y=f(x)$.
Then $b=f(a)$ This means that $a=f^{-1}(b)$, so $D_{1}(b,a)$ is a point on the
graph of the inverse function $f^{-1}$. Now, two points are said to be symmetric with respect to any line if the line is perpendicular to the segm... |
33,430 | <p>Using <kbd>ctrl</kbd><kbd>/</kbd> you can make a fraction. If you have selected something it will appear in the numerator.</p>
<p>Does there exist a shortcut to make the selected text appear in the denominator instead?
If not, is it possible to create a shortcut that does this?</p>
| Pinguin Dirk | 5,274 | <p>As discussed in the comment, it seems you want:</p>
<pre><code>BarChart[Rest /@ data, ChartLabels -> {data[[All, 1]], None},
BarSpacing -> {0, 2}]
</code></pre>
<p>see other options in <a href="http://reference.wolfram.com/mathematica/ref/BarChart.html" rel="noreferrer"><code>BarChart</code></a> to forma... |
33,430 | <p>Using <kbd>ctrl</kbd><kbd>/</kbd> you can make a fraction. If you have selected something it will appear in the numerator.</p>
<p>Does there exist a shortcut to make the selected text appear in the denominator instead?
If not, is it possible to create a shortcut that does this?</p>
| ubpdqn | 1,997 | <p>Is this helpful?</p>
<pre><code>Grid[Partition[BarChart /@ (Transpose[Thread[{#1, ##2}] & /@ data]),
4]]
</code></pre>
<p><img src="https://i.stack.imgur.com/oKUu7.jpg" alt="enter image description here"></p>
<p>You could standardize the plot range.</p>
|
33,430 | <p>Using <kbd>ctrl</kbd><kbd>/</kbd> you can make a fraction. If you have selected something it will appear in the numerator.</p>
<p>Does there exist a shortcut to make the selected text appear in the denominator instead?
If not, is it possible to create a shortcut that does this?</p>
| cormullion | 61 | <p>Out of curiosity I tried this:</p>
<pre><code>DistributionChart[Rest /@ data,
ChartLabels -> {data[[All, 1]]},
ChartElementFunction -> "HistogramDensity",
ChartStyle -> {LightRed, LightGreen, LightBlue},
BarOrigin -> Left]
</code></pre>
<p><img src="https://i.stack.imgur.com/rPlym.png" alt="chart"... |
33,430 | <p>Using <kbd>ctrl</kbd><kbd>/</kbd> you can make a fraction. If you have selected something it will appear in the numerator.</p>
<p>Does there exist a shortcut to make the selected text appear in the denominator instead?
If not, is it possible to create a shortcut that does this?</p>
| VLC | 685 | <p>A solution for <code>PieChart</code> aficionados:</p>
<pre><code>GraphicsGrid[Partition[
Table[PieChart[(Rest /@ data)[[i]],
ChartLabels -> Placed[Range[8], "RadialOutside"],
PlotLabel -> data[[i, 1]]], {i, Length[data[[All, 1]]]}], 4],
ImageSize -> 400]
</code></pre>
<p><img src="https://i.sta... |
1,216,419 | <p>We know that if the given equation were $\quad y = x^3 + ax^2 + bx$, $\quad$ then the derivative would be $3x^2 + 2ax + b$.</p>
<p>Since the given equation is different so the derivative will be: $$2(x^3 + ax^2 + bx)(3x^2 + 2ax + b) \implies 2y(3x^2 + 2ax + b)$$</p>
<p>What does: $$\frac{3x^2 + 2ax + b}{2y}$$
mean... | abel | 9,252 | <p>i think what you want is a $3 \times 3$ matrix $A$ so that $$L(x) = \pmatrix{2x_1 - x_2\\x_2 + x_3\\-3x_1+x_3} = A\pmatrix{x_1\\x_2\\x_3} = \pmatrix{2&-1&0\\0&1&1\\-3&0&1}\pmatrix{x_1\\x_2\\x_3}. $$</p>
|
717,084 | <p>I'm slightly confused as to how </p>
<p>$$\{\emptyset,\{\emptyset,\emptyset\}\} = \{\{\emptyset\},\emptyset,\{\emptyset\}\}$$</p>
<p>are equivalent. I thought two sets were equivalent if and only if "$A$" and "$B$" have exactly the same elements. In this case, we have one element which is in both sets but then two... | wanderer | 59,651 | <p>When you have a set like $\{a,a\}$ then its same as $\{a\}$ so LHS is precisely $\{\phi, \{\phi\}\}$. Similarly, RHS becomes $\{\phi, \{\phi\}\}$ as we are identifying the inner $\{\phi\}$.</p>
|
717,084 | <p>I'm slightly confused as to how </p>
<p>$$\{\emptyset,\{\emptyset,\emptyset\}\} = \{\{\emptyset\},\emptyset,\{\emptyset\}\}$$</p>
<p>are equivalent. I thought two sets were equivalent if and only if "$A$" and "$B$" have exactly the same elements. In this case, we have one element which is in both sets but then two... | kmitov | 84,067 | <p>may be it is better to use that $\{ \emptyset, \emptyset\} = \{\emptyset\} \cup \{\emptyset\}$</p>
|
897,815 | <p>how do I go forward with sketching the graphs of the following two functions?</p>
<p>i) $y(t)=|2+t^3|$</p>
<p>ii) $f(x)=4x+|4x-1|$</p>
<p>thanks in advance!</p>
| user1729 | 10,513 | <p>Double cosets can be used to describe the subgroups of free products and free products with amalgamation (which are important structures in the theory of infinite groups. See the paper <em>The subgroups of a free product of two groups with an amalgamated subgroup</em> by Karrass and Solitar (1970). The description o... |
2,793,380 | <blockquote>
<p>How many ways are there to seat n people in 4 benches so that no bench
is left empty with order?</p>
</blockquote>
<p>Hints from the teacher</p>
<ul>
<li><p>Each bench should have at least 1 person</p></li>
<li><p>This question is similar to distributing different object among n children</p></li>
... | Barry Cipra | 86,747 | <p>A lot depends on exactly what is meant by "with order." If we take it to mean that both the order of the benches and the order in which people sit on each bench matter, then the answer is simple:</p>
<p>$$n!{n-1\choose3}$$</p>
<p>That is, line the people up from left to right, in any of $n!$ ways, then pick $3$ "... |
2,091,761 | <p>If we define a set with $2$ elements in it $S=\{a,b\}$ and a variable "density" $d = 1$ here.</p>
<p>Then if we continue to expand the set with more elements relative to variable $d$ arithmetically, in such a way that:</p>
<p>$$(d=2) \to S= \{a, \frac{a+b}{2},b\}$$
$$(d=3) \to S= \{a, \frac{2a+b}{3}, \frac{a+2b}{3... | Fred | 380,717 | <p>If $a$ and $b$ are rational, then all your sets $S$ from above contain no irrational number !</p>
<p>Your turn !</p>
|
3,475,674 | <p>If the objective function is <span class="math-container">$\min\limits_{x} \sum\limits_{i=1}^n e^{-a_ix_i}$</span>, can I transform the objective into <span class="math-container">$\max\limits_{x} \sum\limits_{i=1}^n a_ix_i$</span>?</p>
| Fimpellizzeri | 173,410 | <p>If one of the <span class="math-container">$a_i$</span>, say <span class="math-container">$a_1$</span>, is <span class="math-container">$0$</span> and all others are positive, then maximizing <span class="math-container">$ \sum\limits_{i=1}^n -a_ix_i$</span> is the same as piling everything onto <span class="math-co... |
818,512 | <p>I have tried to find posts that are related to the question but they end up with the terms like ‘find a distance’. What I want is not to find the distance: I already have the distance, I want something else.</p>
<p>Assume <span class="math-container">$(x_1,y_1)$</span> and <span class="math-container">$(x_2,y_2)$</s... | poolpt | 150,343 | <p>Let $(x_{1},x_{2})$ be a known point on the line</p>
<p>$y=ax+b$</p>
<p>Now you may denote any point on this line by</p>
<p>$(x,ax+b)$</p>
<p>If $d$ is the distance between the points then</p>
<p>$(x-x_{1})^{2}+(ax+b-x_{2})^{2}=d^{2}$</p>
<p>Hence, you have to solve a quadratic equation in x.</p>
|
1,074,740 | <p>We know that torsion-free plus finitely generated <span class="math-container">$\rightarrow$</span> free and that <span class="math-container">$\mathbf{Q}$</span> is torsion-free is easy. </p>
<blockquote>
<p>But how to show <span class="math-container">$\mathbf{Q}$</span> is not finitely generated and not free?<... | k.stm | 42,242 | <p><em>For the non-free part</em>:</p>
<p>Take any two nonzero elements $x, y ∈ ℚ$ and show they satisfy $λx + μy = 0$ for some nonzero $λ, μ ∈ ℤ$, hence any two elements are linearly dependent. Thus, since $ℚ$ is not cyclic, it cannot have a basis.</p>
<p><em>For the non-finitely-generated part</em>:</p>
<p>If $ℚ$ ... |
2,785,160 | <p>Suppose the inequality $\frac {1}{2}-\frac{x^2}{24}<\frac{1-\cos(x)}{x^2}<\frac {1}{2}$ then $\lim_{x\to 0}\frac{1-\cos(x)}{x^2}$ is? I already solved this question by taking limit on the inequalities the answer is $\frac{1}{2}<\lim_{x\to0}\frac{1-\cos(x)}{x^2}<\frac{1}{2}$ but I have a small doubt, as... | user | 505,767 | <p>Recall that by <a href="https://en.wikipedia.org/wiki/Squeeze_theorem" rel="nofollow noreferrer"><strong>squeeze theorem</strong></a> from</p>
<p>$$\frac {1}{2}-\frac{x^2}{24}<\frac{1-\cos(x)}{x^2}<\frac {1}{2}$$</p>
<p>since</p>
<p>$$\frac{1}{2}-\frac{x^2}{24}\to \frac12$$</p>
<p>we can conclude that</p>
... |
2,085,664 | <p>Let $1$ be the multiplicative identity, so that $1\cdot a = a$ (where $a\in \mathbb{F})$. Let $0$ be the additive identity, so that $a+0=a$. Prove that $0\ne 1$. (Here we don't yet know that $0$ and $1$ must be unique, nor do we know that $0\cdot a = 0$).</p>
<p>My approach:</p>
<p>Suppose that $1=0$, then $a+1 = ... | Fnacool | 318,321 | <p>$0\ne 1$ sometimes appears as one of the field axiom, and this is equivalent to requiring the field to have at least two elements.</p>
<p>For the purpose of this exercise, let's assume that this not part of the field axioms, and consider any field with at least two elements, $0$ and $x$. Of course, the field also c... |
438,925 | <p>There are many statements in abstract algebra, often asked by beginners, which are just <em>too good to be true</em>. For example, if <span class="math-container">$N$</span> is a normal subgroup of a group <span class="math-container">$G$</span>, is <span class="math-container">$G/N$</span> isomorphic to a subgroup ... | Benjamin Steinberg | 15,934 | <p>The Nielsen-Schreier theorem that subgroups of free groups are free might have seemed surprising from am algebraic view given the analogue for many other algebraic structures is false. While this is easy to prove topologically, the original algebraic proof is in my view just an algebraic translation of the topologic... |
438,925 | <p>There are many statements in abstract algebra, often asked by beginners, which are just <em>too good to be true</em>. For example, if <span class="math-container">$N$</span> is a normal subgroup of a group <span class="math-container">$G$</span>, is <span class="math-container">$G/N$</span> isomorphic to a subgroup ... | Timothy Chow | 3,106 | <p>The <a href="https://en.wikipedia.org/wiki/Auslander%E2%80%93Buchsbaum_theorem" rel="noreferrer">Auslander–Buchsbaum theorem</a> that every regular local ring is a unique factorization domain.</p>
<p>I should say that the first time I saw this theorem stated, I was not immediately surprised, but that was because I d... |
438,925 | <p>There are many statements in abstract algebra, often asked by beginners, which are just <em>too good to be true</em>. For example, if <span class="math-container">$N$</span> is a normal subgroup of a group <span class="math-container">$G$</span>, is <span class="math-container">$G/N$</span> isomorphic to a subgroup ... | Geoff Robinson | 14,450 | <p>I suppose there is a case for saying that Jordan's theorem on finite complex linear groups might be such a result: there is a function <span class="math-container">$f: \mathbb{N} \to \mathbb{N}$</span> such that for every <span class="math-container">$n \in \mathbb{N}$</span>, every finite subgroup <span class="math... |
2,958,198 | <blockquote>
<p>Suppose you want to distribute <span class="math-container">$15$</span> candies to <span class="math-container">$5$</span> different children.</p>
<p>(a) In how many ways can this be done if no kid receives more than <span class="math-container">$6$</span>
candies?</p>
<p>(b) In how many ways can this b... | dan_fulea | 550,003 | <p>(a) Consider in the ring <span class="math-container">$\Bbb Q[x]/x^{16}$</span> the following product of five equal factors:
<span class="math-container">$$
\left(\ x+x^2+x^3+x^4+\dots+x^{15}+O(x^{16})\ \right)^5\ .
$$</span>
The coefficient <span class="math-container">$C$</span> of <span class="math-container">$x^... |
1,876,086 | <p>The definition of limit says that Let $f(x)$ be a function defined on some open interval that contains the number $a$, except possibly at $a$ itself. Then we say that the limit of $f(x)$ as $x$ approaches $a$ is $L$ If....{the rest of definition is left to make the question easier}. </p>
<p>What does the phrase "ex... | rgnnt | 332,025 | <p>More generally the definition of limit of a function $f$ at a point $a$ requires that $a$ is a limit point for the domain of $f$; see for example Rudin's book third edition definition 4.1.</p>
<p>When $a$ is a limit point for the domain of a function $f$, you are sure that it's always possible evaluate $f(x)$ when ... |
1,876,086 | <p>The definition of limit says that Let $f(x)$ be a function defined on some open interval that contains the number $a$, except possibly at $a$ itself. Then we say that the limit of $f(x)$ as $x$ approaches $a$ is $L$ If....{the rest of definition is left to make the question easier}. </p>
<p>What does the phrase "ex... | Community | -1 | <p>Here's an answer at more of a calculus level and not so much a real analysis level.</p>
<blockquote>
<p>Let $f(x)$ be a function defined on some open interval that contains the number $a$, except possibly at $a$ itself.</p>
</blockquote>
<p>"except possibly at $a$ itself" means the function may not actually be d... |
2,136,859 | <p>Is there a rational solution for the following equation?
<span class="math-container">$$\tan (\pi x)=y\\y\neq-1,0,1$$</span></p>
<p>I guess there is none, but I have no idea how to solve/prove it.</p>
<p>EDIT: I think also that if <span class="math-container">$y$</span> is rational, then <span class="math-containe... | Curtis McMullen | 558,532 | <p>Here is a short elementary argument. Suppose <span class="math-container">$\tan(\pi x) = y$</span> with <span class="math-container">$x$</span> and <span class="math-container">$y$</span> rational. Write <span class="math-container">$x =p/q$</span> with <span class="math-container">$\gcd(p,q)=1$</span>. Then <spa... |
1,834,837 | <p>I had the following two problems:</p>
<blockquote>
<p>Find a counterexample for $f_*(A \cap B) \supseteq f_*(A) \cap f_*(B)$ and $ f_*(A-B) \subseteq f_*(A) -f_*(B).$ Where $f_*(X)$ is the image of $X$ under $f$ for some function $f:A \rightarrow B $ and some subset $X \subseteq A$.</p>
</blockquote>
<p>I came w... | Stefan Mesken | 217,623 | <p>Your answers are correct and as you already stated, there is no general rule to produce counterexamples to the converse of a given result. However, it is often helpful to look at the proof of the correct result and study closely how the premises come into play. At every step where they're used, you may then try to p... |
3,380,849 | <p>Is the equation I wrote in the title true for positive integers <span class="math-container">$x,y$</span>? I checked some cases and it seems to hold, but how do I prove it? I am trying to solve another problem and it turns out that if this equation holds, the problem is solved.</p>
| Surajit | 528,430 | <p><strong>Hint:</strong> <span class="math-container">$\\lcm(x,y)=\frac{xy}{\gcd(x,y)}$</span> and <span class="math-container">$\gcd(x,x+y)=\gcd(x,y)$</span>.</p>
|
3,380,849 | <p>Is the equation I wrote in the title true for positive integers <span class="math-container">$x,y$</span>? I checked some cases and it seems to hold, but how do I prove it? I am trying to solve another problem and it turns out that if this equation holds, the problem is solved.</p>
| Stinking Bishop | 700,480 | <p>It is true:</p>
<p><span class="math-container">$$\begin{array}{rcl}(x+y)\cdot \text{lcm}(x,y)&=&(x+y)\cfrac{xy}{\gcd(x,y)}\\&=&y\cfrac{x(x+y)}{\gcd(x,x+y)}\\&=&y\cdot \text{lcm}(x,x+y)\end{array}$$</span></p>
<p>I've used two facts here:</p>
<p><span class="math-container">$$\text{lcm}(x,... |
2,719,542 | <blockquote>
<p>Suppose that
$$\int_{-1}^1 f(x)dx=5$$
$$\int_{1}^4 f(x)dx=-2$$
$$\int_{-1}^4 h(x)dx=7$$
Find the value of
$$\int_{-1}^4 (2f(x)+3h(x))dx$$</p>
</blockquote>
<p>I understand how to find definite and indefinite integrals, but I'm not entirely sure how to even begin this problem.</p>
| achille hui | 59,379 | <p>There is no need of any complicated algebra. Following is a geometric way to get the answer. One advantage of this approach is you don't need to assume the largest rectangle is axis aligned with the ellipse.</p>
<p>Given any circle, it is well known the largest quadrilateral inscribed in it is a square. Furthermore... |
1,168,402 | <p>Solve DE: $$y'' + 2y' + 5y = x + 4$$</p>
<p>I have the correct general solution
$$y(x) = c_1\,e^{-x}\cos(2x) + c_2\,e^{-x}\sin(2x)\ .$$</p>
<p>So I take my 'guess' , take a couple of derivatives, and plug in for the equation $$2A +5(Ax + B) = x + 4$$</p>
<p>But at this point I'm stuck on solving for the constant... | mvw | 86,776 | <p>The ansatz for $y_p = A x + B$ put in the DE gives
$2A + 5(Ax + B) = x + 4$ where you compare coefficients for $x^k$, here $x^1=x$ and $x^0=1$, on both sides of the expanded equation. Finally $y = y_h + y_p$.</p>
|
2,810,410 | <p>Define $B: \mathbb{R^3} × \mathbb{R^3} \to \mathbb{R}$ with $B((x_1,x_2,x_3),(y_1,y_2,y_3)) := -2x_1y_1-x_2y_3-x_3y_2$.</p>
<p>How to check if vectors $v \in \mathbb{R^3}, v \neq 0$ exist such that $B(v,v) = 0$?</p>
<p>I don't know how to start here.</p>
<p>Also, how to find a basis of $\mathbb{R^3}$ such that th... | Mike Earnest | 177,399 | <p>Hint: Letting $V_1,V_2,\dots$ be a sequence of iid random variables distributed uniformly on $[0,1]$, show that $X_n$ has the same distribution as $V_1\cdot V_2\cdot\ldots \cdot V_n$. Next, find the distribution of $\log X_n$, which is a <em>sum</em> of the iid variables $\log V_i$ (what distribution does $\log V_i$... |
993,132 | <p>I have been looking into this question :
we have two surfaces :</p>
<p>$$\big\{(x,y,z)\in \mathbb{R}^3 \mid\;\; S_1\colon\;\; x+z=1 ,\;\; S_2\colon\;\; x^2+y^2=1 \big\}$$</p>
<p>we need to draw or describe the "shape" that we get .
I tried to solve it by drawing the two surfaces and imagining the intersection whic... | Martín-Blas Pérez Pinilla | 98,199 | <p>The surfaces are a vertical cylinder and a oblique plane. The intersection is obviously a ellipse. Do you want a <em>parametrization</em>? Project on the plane $XY$, parametrize the projection and use the other equqtion:
$$x=\cos t,$$
$$y=\sin t,$$
$$z= 1 - \cos t,$$
$(t\in [0,2\pi])$.</p>
|
1,613,886 | <p>I was given a problem to minimise </p>
<p>$$[(x-y)^2+(12+\sqrt{1-x^2} -\sqrt{4y})^2]$$</p>
<p>Where x,y are real, I have managed to solve it, but it took a lot of time and effort, can anyone provide a short way?</p>
| Nikunj | 287,774 | <p>If it's unclear , ask for any clarifications.
<img src="https://i.stack.imgur.com/HEQvh.jpg" alt="enter image description here"></p>
|
107,882 | <p>Can someone recommend a good basic book on Geometry? Let me be more specific on what I am looking for. I'd like a book that starts with Euclid's definitions and postulates and goes on from there to prove thereoms about triangles, circles and other plane shapes. I'm not interested (at this time) in a book that tie... | bgins | 20,321 | <p>Searching for Geometry textbooks between about 1970 & 1982
in <a href="http://mathcurriculumcenter.org/" rel="noreferrer">one possible database</a> (probably not very complete),
I found:</p>
<ul>
<li><a href="http://books.google.ch/books?id=LPqlAAAACAAJ&dq=geometry+0395182948&hl=en&sa=X" rel="norefe... |
107,882 | <p>Can someone recommend a good basic book on Geometry? Let me be more specific on what I am looking for. I'd like a book that starts with Euclid's definitions and postulates and goes on from there to prove thereoms about triangles, circles and other plane shapes. I'm not interested (at this time) in a book that tie... | Nebo Alex | 218,007 | <p>You can checkout <strong>Euclid's Elements Of Geometry</strong> .</p>
<p>The Greek text of J.L. Heiberg (1883–1885) from Euclidis Elementa, edidit et Latine interpretatus est I.L. Heiberg, in aedibus B.G. Teubneri, 1883–1885
edited, and provided with a modern English translation, by
Richard Fitzpatrick.
I haven't r... |
3,090,052 | <p>My question concerns the answer to exercise 1.3:</p>
<blockquote>
<p>Given a partition <span class="math-container">$P$</span> on a set <span class="math-container">$S$</span>, show how to define a relation <span class="math-container">$\sim$</span> on <span class="math-container">$S$</span> such that <span class... | Felix Marin | 85,343 | <p><span class="math-container">$\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
\newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
\newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
\newcommand{\dd}{\mathrm{d}}
\newcommand{\ds}[1]{\displaystyle{#1}}
\newcommand{\exp... |
3,505,675 | <p>Proposition: Let <span class="math-container">$(X, d)$</span> be a metric space, and <span class="math-container">$S$</span> a subset of <span class="math-container">$X$</span>. If <span class="math-container">$B_d(x, r) \cap S \neq \emptyset$</span> for all <span class="math-container">$r >0$</span>, then there ... | Barry Cipra | 86,747 | <p>The author of the guide is trying to teach some elements of compositional style, emphasizing the idea that mathematics should be presented in a way that is readable as sensible English sentences. From this point of view, the two-column display reads as an alternation of declaratives and imperatives (i.e., statements... |
552,307 | <blockquote>
<p>If A $\cap$ B $\cap$ C = $\emptyset$, then the sum principle applies so |A $\cup$ B $\cup$ C| = |A|+|B|+|C|.</p>
</blockquote>
<p>I think it would be true since there is nothing in common among A,B and C, but just wondering if there is any exceptions to this problem so it would be false?</p>
| Silverfish | 72,968 | <blockquote>
<p>Let A = {1,2}, B={2,3} and C={3,1} so |A| = |B| = |C| = 2. Then A $\cap$ B $\cap$ C = $\emptyset$, but $$A \cup B \cup C = \{1,2,3\} \implies |A \cup B \cup C|=3 \not= |A| + |B| + |C|$$</p>
</blockquote>
<p>When you apply <a href="http://en.wikipedia.org/wiki/Inclusion%E2%80%93exclusion_principle" re... |
4,238,492 | <p>A practical example lead me to believe that a geographical projection, such as the <a href="https://en.wikipedia.org/wiki/Mercator_projection" rel="nofollow noreferrer">Mercator projection</a>, is an <a href="https://en.wikipedia.org/wiki/Affine_transformation" rel="nofollow noreferrer">affine transformation</a>.</p... | PrincessEev | 597,568 | <p><span class="math-container">$
\newcommand{\e}{\varepsilon}
\newcommand{\a}{\alpha}
$</span>
One useful characterization of the infimum is the <span class="math-container">$\varepsilon$</span>-characterization. For <span class="math-container">$A \subseteq \Bbb R$</span>, and <span class="math-container">$\a$</span>... |
572,307 | <p>I am having trouble proving the following statement: </p>
<blockquote>
<p>Prove that for all integers $m$ and $n$, if $d$ is a common divisor of $m$ and $n$ (but $d$ is not necessarily the GCD) then $d$ is a common divisor of $n$ and $m - n$.</p>
</blockquote>
<p>I've noticed that for any integers $m,n,d$ that $... | hejseb | 70,393 | <p>If $d$ divides $m$, then $m=m_1d$ for some integer $m_1$. Similarly, if $d$ divides $n$, then $n=n_1d$ for some integer $n_1$. And hence
$$
m-n=d(m_1-n_1).
$$</p>
|
694,253 | <p>How might I find tail probabilities (pr X>x), or a reasonable approximation, for a variable that is the sum of independent Laplace random variables? </p>
| Batman | 127,428 | <p>A simple bound that is often useful is to use the Chernoff bound:</p>
<p>From Markov's inequality, we know for a non-negative r.v. <span class="math-container">$X$</span>, <span class="math-container">$P(X \geq c) \leq \frac{E[X]}c$</span>. Now, introduce a non-negative parameter <span class="math-container">$t$</sp... |
47,011 | <p>This question was posed originally on <a href="https://math.stackexchange.com/questions/10990/uses-of-divergent-series-and-their-summation-values-in-mathematics">MSE</a>, I put it here because I didn't receive the answer(s) I wished to see.</p>
<p>Dear MO-Community,</p>
<p>When I was trying to find closed-form rep... | Gerald Edgar | 454 | <p>RULES ... don't just use divergent series to get an answer unless there is some additional work to show it is meaningful. Indeed, don't even use rearrangement of conditionally convergent series and expect to get something meaningful (again) unless there is some additional work.</p>
<p>AN EXAMPLE ... <strong>Fourie... |
287,154 | <p>Note: I asked the question below last week on MathSE but received no answer. </p>
<p>Background:</p>
<p>I have read the claim that perverse sheaves behave more like sheaves than like complexes of sheaves. This refers to the fact that they can be glued. </p>
<p>For instance, suppose that $X$ is a complex analyti... | gdb | 115,211 | <p>This doesn't quite answer your question, but it might be useful. </p>
<p>First of all, note that it is extremely unbelievable (please, correct me if I am wrong) that you can glue objects of (bounded) derived category, if you cannot glue morphisms. One reason is that you usually construct gluing "by induction on num... |
4,091,264 | <p>Here is my attempt to prove this implication, but I got stuck.</p>
<p><span class="math-container">$$
(p\implies q)\land(r\implies s)\land(p\lor r)\implies (q\lor s)\\
(\neg p\lor q)\land(p\lor r)\land(r\implies s)\\
[(\neg p\lor q)\land p]\lor[(\neg p\lor q)\land r]\land(r\implies s)\\
[(\neg p\land p)\lor(q\land p... | Shaun | 104,041 | <p>One way is the <a href="https://en.m.wikipedia.org/wiki/Method_of_analytic_tableaux" rel="nofollow noreferrer">method of analytic tableaux</a>.</p>
<p>You start with the negation of your formula then apply some contradiction-hunting rules to deduce that it is impossible, like so:</p>
<p><img src="https://i.stack.img... |
4,091,264 | <p>Here is my attempt to prove this implication, but I got stuck.</p>
<p><span class="math-container">$$
(p\implies q)\land(r\implies s)\land(p\lor r)\implies (q\lor s)\\
(\neg p\lor q)\land(p\lor r)\land(r\implies s)\\
[(\neg p\lor q)\land p]\lor[(\neg p\lor q)\land r]\land(r\implies s)\\
[(\neg p\land p)\lor(q\land p... | David J. Webb | 884,214 | <p>It suffices to show that when we have the hypotheses and <span class="math-container">$\lnot q$</span>, we can derive <span class="math-container">$s$</span>.</p>
<p>From contraposition, the first implication in our hypotheses gives us <span class="math-container">$\lnot p$</span>. From the third we must then have <... |
1,712,089 | <p>Probability for a disease is $0.05$.The probability that a diagnosis device will give positive result if the person has the disease is $0.99$ and vice versa. </p>
<p>a- If test is positive what is the probability that a person has the disease?</p>
<p>b- If test is applied to 2 persons and both show positive what i... | Evariste | 239,682 | <p>One person is chosen randomly for the test. Since the probability to be ill is 0.05, 5% of the population is ill, assuming it is big enough to neglect statistical dispersion. So the probability to pick a healthy individual is 0.95 and the probability to pick an ill person is 0.05.</p>
<p>Among the diseased, 99% of ... |
1,280,882 | <p>This may be a silly question, but one that I am confused about nonetheless.</p>
<p>With regards to the compound trig identities such as $\cos(A+B)=\cos A\cos B - \sin A\sin B$ etc., I'd like to know why they are used. What's the purpose? Surely, one would ask themselves that if we can just add $A$ and $B$ together ... | Gregory Grant | 217,398 | <p>Here's an example of where it can be useful. Let's use it on $\cos(x+\pi/2)$. The formula says $\cos(x+\pi/2)=\cos(x)\cos(\pi/2)-\sin(x)\sin(\pi/2)$ and since $\cos(\pi/2)=0$ and $\sin(\pi/2)=1$ this implies $\cos(x+\pi/2)=-\sin(x)$ The sum formulas can be used to prove all kinds of useful identities like that.</... |
2,233,369 | <p>If there exists a homeomorphism $f$ between the closed unit interval and some cartesian product $A\times B$, either $A$ is a singleton or $B$ is a singleton.</p>
<p>The proof I have argues as follows:</p>
<p>Since $[0,1]$ is connected, $A\times B$ is, so $A$ and $B$ are as well. If $a_1,a_2 \in A$ and $b_1,b_2\in ... | freakish | 340,986 | <blockquote>
<p>I don't understand why the $f^{−1}(\{a_1\}\times B)$ and so on are closed?</p>
</blockquote>
<p>$\{a_1\}\times B$ is closed in $X\times Y$ by the definition of product topology and since $f$ is continous then $f^{−1}(\{a_1\}\times B)$ is closed in $I$.</p>
<p>Note that $\{a_1\}\subseteq A$ is closed... |
264,061 | <p>I need help with the following problem. Suppose $Z=N(0,s)$ i.e. normally distributed random variable with standard deviation $\sqrt{s}$. I need to calculate $E[Z^2]$. My attempt is to do something like
\begin{align}
E[Z^2]=&\int_0^{+\infty} y \cdot Pr(Z^2=y)dy\\
=& \int_0^{+\infty}y\frac{1}{\sqrt{2\pi s}}e^{... | Ron Gordon | 53,268 | <p>The answer is $s = \sigma^2$. The integral you want to evaluate is </p>
<p>$$E[Z^2] = \frac{1}{\sqrt{2 \pi} \sigma} \int_{-\infty}^{\infty} dz \: z^2 \exp{(-\frac{z^2}{2 \sigma^2})}$$</p>
|
1,149,602 | <p>The parametric curve $r=(-2t^2+8t-2,cos({\pi}t),t^3-28t)$ crosses itself at one and only one point. The point is $(x,y,z)$. I found $t=-2$ to be the answer and $(x,y,z)=(-26,-1,48)$ to be the correct answer. </p>
<p>However it asks for the acute angle between the two tangent lines to the curve at the crossing point... | abel | 9,252 | <p>suppose the trajectory crosses itself at $t = a, t = b \neq a,$ the we have $r(a)= r(b).$ there are three simultaneous equation to solve: the first one $$-2a^2 + 8a - 2 = -2b^2 +8b -2$$ implies $b + a = 4.$ if i substitute in the $z$ component i get $$a^3 - 28a = (4-a)^3 - 28(4-a).$$ this equation has thre roots $... |
2,132,994 | <p>I need to prove that absolute value of any real number is greater than or equal to that real number, where $|a| = a ; a\ge0 , |a| = -a ; a<0 $</p>
<p>I came across this on real analysis.
I need this proven Filed and Order Axioms and basic definitions.</p>
| fleablood | 280,126 | <p>Definition of absolute value is $|a|=a $ if $a \ge 0$. $|a|=-a $ if $a <0$.</p>
<p>If $a\ge 0$ then $a \le a =|a|$.</p>
<p>If $a < 0$ then we must prove $a \le -a =|a|$</p>
<p>$a < 0$. By axiom $x <y\implies x+w <y+w $</p>
<p>That should do it for you. (Let $x=a;y=0; $ and $w =???? $)</p>
|
3,000,052 | <p>Given <span class="math-container">$Z_n=\arg{\frac{i^n}{n}}$</span>, how do I show that it has no limit?</p>
| Przemysław Scherwentke | 72,361 | <p>HINT: Observe that <span class="math-container">$Z_n$</span> takes only four values.</p>
|
1,556,609 | <p>Let $C$ be the curve of intersection of the two surfaces $x+y=2 , x^2+y^2+z^2=2(x+y)$ . The curve is to be traversed in clockwise direction as viewed from the origin . The what is the value of $\int_Cydx+zdy+xdz$ ? I am not even able to parametrize the curve of intersection . Please help . Thanks in advance </p>
| Kuifje | 273,220 | <p>Note that your integral is equivalent to:</p>
<p>$$
\int_C \vec{F} \cdot d\vec{r},
$$</p>
<p>with $\vec{F}=(y,z,x)$.</p>
<p>Now, lets have a look at $C$: it is the intersection between the plane $x+y=2$, and the sphere $(x-1)^2+(y-1)^2+z^2=2$. In other words $C$ is an ellipse. Most importantly, it is a <strong>cl... |
4,501,398 | <p>First of all forgive my very poor and not to scale drawing. Also for the not so good looking maths formatting</p>
<p>Essentially I am looking for the area of the shaded part.
This is what I've gotten so far</p>
<p>Area of larger circle with radius being <span class="math-container">$20+w$</span> minus area of inner ... | Shinrin-Yoku | 789,929 | <p><span class="math-container">$\{0\}$</span> is a finite set <br>
<span class="math-container">$\{0,1\}$</span> is a finite set<br>
<span class="math-container">$\{0,1,2\}$</span> is finite set<br> <span class="math-container">$
\vdots$</span></p>
<p>Repeating to infinity gives us that the set of all naturals is a fi... |
127,186 | <p>Like my previous question, I'll pose this one too with an array.</p>
<p>$1^r, 3^r, 5^r, 7^r, 9^r$ (all odd $r$th powers)</p>
<p>That's array 1. And array 2;</p>
<p>$2^r, 4^r, 6^r, 8^r, 10^r$ (all even $r$th powers)</p>
<p>Let's take the sum of random $k$ integers from each array, where $r>2$ (to eliminate Pyt... | Douglas S. Stones | 139 | <p>There are $O(n)$ vertices in $G$, each of which can be an endpoint of at most $10^1+10^2+10^3+\cdots+10^{19}=O(1)$ walks of lengths between $1$ and $19$.</p>
<p>Hence, brute-force checking all of these $O(n)$ walks (e.g. via depth-first search or breadth-first search) will run in $O(n)$ time (albeit with a horribly... |
2,654,606 | <p>In Chapter 7 ("Inverse limits and direct limits"), subchapter 4 ("Conditions for an inverse limit to be non-empty"), Bourbaki lets $(E_\alpha)_{\alpha \in I}$ be a projective system of sets with connecting maps $(f_{\alpha \beta}) _{\alpha, \beta \in I}$, and for each $\alpha \in I$ he lets $\mathfrak S_\alpha$ be a... | Noah Schweber | 28,111 | <p>Remember that the intersection of the empty family consists of <em>everything</em>: $$\bigcap\emptyset=\{x: \forall a\in\emptyset(x\in a)\}=\{x: x=x\}.$$ Now of course there isn't actually a universal set, so this is an abuse of terminology. When we speak of "the intersection of the empty family," we really mean thi... |
353,374 | <p>If we throw three dice at the same time and see a sum of numbers. What number will have the greatest probability. I think that that number is 11 but I am not positive. Any help will be appreciated. </p>
| Metin Y. | 49,793 | <p><strong>Hint</strong>: Consider only one die and try to think of which number appears the most as a sum of two numbers of the die. It is $7=1+6=2+5=3+4$. Agree? You try to pick the kind of "middle ones" to get your sum.</p>
|
3,590,250 | <p>For <span class="math-container">$r>0$</span>, let be
<span class="math-container">$$I(r)=\int_{\gamma_r}\frac{e^{iz}}{z}dz$$</span>
where <span class="math-container">$\gamma_r:[0,\pi]\to\mathbb{C}, \gamma_r(t)=re^{it}$</span>. Prove that <span class="math-container">$\lim_{r\to\infty}I(r)=0$</span>.</p>
<p>I'v... | stokes-line | 756,028 | <p>It is a particular case of Jordan's <a href="https://en.wikipedia.org/wiki/Jordan%27s_lemma" rel="nofollow noreferrer">lemma</a>.
The proof goes essentially as follows. First, you split the integration region into two halves <span class="math-container">$[0,\pi/2]$</span> and <span class="math-container">$[\pi/2,\pi... |
3,354,466 | <p>my question is:</p>
<p>Given a line <strong>S</strong>: <span class="math-container">$$
\left\{
\begin{array}{c}
x+y-1=0 \\
y+3z-2=0 \\
\end{array}
\right.
$$</span>
I need to determine the equation of a sphere having the center on line <strong>S</strong> and tangent to the plane <span class="math-container">$... | Community | -1 | <p>Take any point on the line and consider the sphere centered on this point, with a radius equal to <span class="math-container">$|z|$</span>. It fulfills the requirements.</p>
|
1,525,340 | <p>I have a fairly simply question which I am not sure about. A 3 digits number is being chosen by random (100-999). What is the probability of getting a number with two identical digits ? (like 101). Thank you !</p>
| Nate 8 | 226,768 | <p>The best way to approach these problems is to make lists of what could happen, and pick numbers one by one.</p>
<p>So the first number must be 1-9 The second number must be 0-9. (No restrictions so far.)</p>
<p>But we have to pick the third number carefully. If the first two numbers matched, we must pick a differe... |
419,091 | <blockquote>
<p><span class="math-container">$G$</span> is an infinite group.</p>
<ol>
<li><p>Is it necessary true that there exists a subgroup <span class="math-container">$H$</span> of <span class="math-container">$G$</span> and <span class="math-container">$H$</span> is maximal ?</p>
</li>
<li><p>Is it possible that... | Seirios | 36,434 | <p>In the same way, you have $$\mathfrak{S}_2 \subsetneq \mathfrak{S}_3 \subsetneq \cdots \subsetneq \mathfrak{S}_n \subsetneq \cdots \subsetneq \mathfrak{S}_{\infty}$$ where $\mathfrak{S}_{\infty}$ is the set of bijections $\mathbb{N}\to \mathbb{N}$ fixing all but finitely many numbers.</p>
|
558,156 | <p>Prove that $\binom{n}{r} + \binom{n}{r+1} = \binom{n+1}{r+1} $</p>
<p>Thanks in advance, my professor asked us to this a couple weeks ago, but I was enable to get to the right answer. </p>
<p>Good luck!</p>
<p>Here is what I got up to;</p>
<p>$\frac{(n+1)!}{(n-r)!(r+1)!} = \frac{(n)!}{(r)!(n-r)!} + \frac{(n)!}{... | Elias Costa | 19,266 | <p>Even though it seems a little far-fetched I will use the Binomial Theorem. The definition of number $\binom{n}{r}$ has a reason to come to exist in the development of $(x + y)^n$. So I think the natural and instructive. For all $x,y\in\mathbb{R}$ we have,
\begin{array}{rrl}
\hspace{2cm}&( x+y)^{n+1}= & ... |
1,893,356 | <p>This is what I came up with so far:</p>
<p>Inductive step: assume $2^n > n^4$.
Need to prove $2^{n+1} > (n+1)^4$
$$
2^{n+1} = 2 \cdot 2^n > 2 \cdot n^4\\
(2 \cdot n^4)^{1/4} = (2)^{1/4} \cdot n > n+1 \implies 2n^4 > (n+1)^4 \implies 2^n > (n+1)^4
$$</p>
<p>Is there a better way to solve this prob... | ajotatxe | 132,456 | <p>There is another inductive way, I don't know if better or worse:</p>
<p>Our goal is to prove that $2^{n+1}>(n+1)^4$, assuming that $2^n>n^4$ and, as noted, $n\ge 17$. Let's estimate $2^{n+1}-(n+1)^4$:</p>
<p>$$2^{n+1}-(n+1)^4=\big(2^n-[(n+1)^4-n^4]\big)+(2^n-n^4)>(2^n-n^4)+(2^n-n^4)>0$$</p>
<p>To show... |
1,314,346 | <p>I'm learning about generating functions, and one part that I am struggling with is the logic behind rearranging summations (particularly double summations).</p>
<p>I'll illustrate an example:</p>
<p>Using the Lagrangean Inversion theorem, I get that the $z^n$ coefficient is given by:
$$S_n=\frac{1}{n}[u^{n-1}]\le... | Adelafif | 229,367 | <p>the triangle is equilateral since if two angles are congruent then the corresponding sides are congruent in any triangle in a Hilbert's space geometry. See Hartshorne: Geometry</p>
|
113,295 | <p>Suppose $x_1, x_2$ are IDD random variables uniformly distributed on the interval $(0,1)$. What is the pdf of the quotient $x_2 / x_1$?</p>
| Sasha | 11,069 | <p>Because $X_2$ is positive with almost surely, cumulative distribution function for $Z=X_1/X_2$ is
$$
F_Z(z) = \mathbb{P}(Z \leqslant z) = \mathbb{P}(X_1 \leqslant z X_2) = \mathbb{E}_{X_2}\left( \mathbb{P}(X_1 \leqslant z X_2 | X_2)\right) = \mathbb{E}_{X_2}\left( F_{X_1}(z X_2)\right)
$$
Clearly $F_Z(z)=0$ for... |
20,714 | <p>The first three expressions evaluate as expected and the polynomial is displayed in what I would call "textbook" form. The last expression, however, switches the order of terms. Mathematica employs this change for two-term polynomials if it results in getting rid of the leading negative sign (at least that is the be... | Mr.Wizard | 121 | <p>I hope there is a better way but here is something to build upon:</p>
<pre><code>TraditionalForm @ Row[MonomialList@#, "+"] & /@
{x^2 + x + 5, -x^2 + x + 5, x^2 + x, -x^2 + x}
</code></pre>
<p><img src="https://i.stack.imgur.com/nXU4z.png" alt="Mathematica graphics"></p>
<hr>
<p>Jens pointed out a bug in m... |
66,474 | <p>I am trying to solve a 1st order non-linear ODE</p>
<pre><code>W[y]*W'[y] + W[y]*v + Fnum == 0 /. v -> 10
Fnum = 0.05 - 1.66667 y + (270.27 y)/(1 + 270.27 y) - (660. y)/(1 + 1666.67 y)
</code></pre>
<p>The function Fnum has three zeros. two of them are at y=StartY and y=EndY.</p>
<pre><code>In[1]:= StartY = 0... | bbgodfrey | 1,063 | <p>In an earlier comment, I observed that <code>NDSolve</code> found the equation to be stiff, even when the calculation was begun a short distance <code>d</code> from <code>StartY</code>. Because the third root of <code>Fnum</code> is <code>0.000797339</code>, <code>d</code> must be quite small, say, 0.0001. However,... |
760,098 | <blockquote>
<p>By drawing graph,or otherwise,find the <strong>number of roots</strong> of the equation
$x+2 \tan(x)= \pi/2$<br>
lying between $0$ and $2\pi$, and find the <strong>approximate value of the largest root</strong>.</p>
</blockquote>
<p>I found <strong>3</strong> roots by drawing a rough graph ,but... | Claude Leibovici | 82,404 | <p>Drawing the graph, you identify that the three roots are located close to $x=0.5$, $x=2.5$ and $x=5.0$. </p>
<p>The solutions of equations such as the one you post do not have solutions which can be expressed using elemental functions (I do not thing that, even using complex functions, there is any analytical solut... |
2,904,878 | <p>I want to prove that
let S be the set on which group G operates. Let H ={ g∈G | g.s=s for all s∈S} prove that H is normal subgroup of G.
This group action gives an homomorphism whose kernal is H.
Then it follows directly from statement that H is normal.</p>
<p><strong>How to prove that if the subgroup is the kerne... | Kavi Rama Murthy | 142,385 | <p>Let $a=-1,b=1$, $f(x)=x, g(x)=x+2$. Then $g(x) >0$ and $\int_a^{b} f(x) \, dx=0, I=\frac 2 3$. So there is no such point $c$.</p>
|
8,891 | <p>This is not a question. Just a request.</p>
<p>This Issue has been discussed previously and I think this can be implemented technically as well.</p>
<p>Writing title all in latex disables the options like "Open in New Tab" and other options in browsers. It may be user's own habit but I terribly hate it when the op... | user642796 | 8,348 | <p>One should first note that <a href="http://blog.stackoverflow.com/2011/04/stack-exchange-partners-with-mathjax/">only a few StackExchange sites have MathJax capabilities</a>. Add in the fact that such questions appear only rarely and it is probably unlikely that SE will implement a change to require titles to have ... |
1,440,988 | <p>Show that the set $ \left\{\dfrac{1}{x^2-1}\mid x\in(0,1)\right\} $ is not bounded.</p>
<p>We should assume that it is bounded, then try to prove the opposite, but I don't know where to start. </p>
| Raj | 115,172 | <p>$\frac{1}{x^2 - 1} < 0$ for all $x \in (0, 1)$. Notice that the value of the expression becomes more and more negative as $x$ approaches 1. So, let $M < 0$ be arbitrary. Then all you need to do to show that $\frac{1}{x^2 - 1}$ is unbounded is find a special value of $x \in (0, 1)$ such that,</p>
<p>$$
\frac{1... |
1,894,699 | <p>$d(x,S) = \inf_{s \subset \mathbb{R}}\{|x-s|: s \in S\}, x\in \mathbb{R}$</p>
<p>I did notice this question was asked before, but most people were asking for tips or completely solve it for them. I want to get critique on my work as I seem to be the only to have attempted it.</p>
<p>How I went about it:</p>
<p>If... | Community | -1 | <p>Let $x=\tan A, y = \tan B, z = \tan C$. Then
\begin{align*}
\tan(A+B+C) = \frac{\tan A + \tan B + \tan C - \tan A \tan B \tan C}{1-\tan A \tan B -\tan B \tan C-\tan C \tan A}
\end{align*}
Hence $A+B+C $ is an odd multiple of $\frac{\pi}{2}$. We need to prove
\begin{align*}
\tan 2A + \tan 2B + \tan 2C = \tan 2A \... |
766,629 | <p>Suppose I am given two finite groups $G$ and $H$ (not too large: let's say their orders are around $10000$ and $100$ respectively, and the order of $H$ divides the order of $G$). These may be represented as groups of permutations with known, fairly small, sets of generators. I would like to find, if possible, a su... | Olexandr Konovalov | 70,316 | <p>To complement Derek's answer with an example in <a href="http://www.gap-system.org/">GAP</a>, the first three blocks of commands from above look almost identical. Note that <code>H</code> is taken in the polycyclic representation (if one need an fp-group, in GAP the command would be <code>Image(IsomorphismFrGroup(H)... |
172,432 | <p>Jyrki Lahtonen has suggested I write a blog post relating binary quadratic forms to quadratic field class numbers, <a href="https://math.stackexchange.com/questions/209512/binary-quadratic-forms-over-z-and-class-numbers-of-quadratic-%EF%AC%81elds/209543#comment1727526_209543">https://math.stackexchange.com/questions... | Jeremy Rouse | 48,142 | <p>It seems to me that what Buell says about the narrow class group is not quite right (it's hard for me to say, as I don't have a copy of it). Magma tells me that in $\mathbb{Q}(\sqrt{210})$, the narrow class group is $(\mathbb{Z}/2\mathbb{Z})^{3}$ and the ideal class group is $(\mathbb{Z}/2\mathbb{Z})^{2}$. The squar... |
621,210 | <p>As it is clear from the title, what is the cardinality of the set $\{ (x,y) \in \Bbb{R}^2 \; | \; y > x > 0 , x^x = y^y \}$?</p>
| Giulio Bresciani | 118,028 | <p>Use $x=z^{\frac z{1-z}}$ and $y=z^{\frac 1{1-z}}$ for $0<z<1$. We have $y=zx$, and</p>
<p>$$y^y=z^{\frac y{z-1}}=z^{\frac {z}{z-1}x}=x^x$$</p>
|
1,113,516 | <p>I learnt that $(\mathbb{R},\times) < (\mathbb{C},\times)$, Which means the first is a subgroup of the second one. But in the first group inequality is defined, while it's not in the latter. This got me thinking, is there anything about a group which tells you which symbols ($=,\neq,<,\geq,\cdots$) are defined ... | Pp.. | 203,995 | <p>Let $X_i:=[a_i,b_i]^T$ then $$X_{i+1}=AX_i$$ where $$A:=\begin{bmatrix}\delta&\lambda_1\\\lambda_2&\delta\end{bmatrix}$$</p>
<p>Then it is clear that $$X_i=A^{i}X_0$$</p>
<p>To get a nice formula for $A^i$ first write it as $P^{-1}JP$, where $J$ is its Jordan form. Notice that if $\lambda_1\lambda_2\neq0$ ... |
4,445,342 | <p>I am working on SL Parsonson's <em>Pure Mathematics</em> and I haven't been able to solve this problem:</p>
<p><span class="math-container">$n^2$</span> balls, of which <span class="math-container">$n$</span> are black and the rest white, are distributed at random into <span class="math-container">$n$</span> bags, s... | aschepler | 2,236 | <p>I'm imagining instead of bags, we have an <span class="math-container">$n \times n$</span> grid of dimples where we can place balls. Then we'll consider each column of the grid to correspond to a bag.</p>
<p>You're correct that the number of ways to place <span class="math-container">$n$</span> (identical) black bal... |
1,914,179 | <p>We have three balls in every move we can swap two of the balls.Prove that after an odd number of moves the permution of the balls are not same as they was first.</p>
<p><strong>My attempt</strong>:I cant do anything special but I find out that at last one of the balls is in its first place but the two other's are s... | PSPACEhard | 140,280 | <p><strong>A Simple Proof Based on Bipartite Graph.</strong></p>
<p>Let the balls be $1$, $2$ and $3$. There are totally $6$ states, namely, $123$, $132$, $213$, $231$, $312$ and $321$. Draw a node for each state and draw a line between two nodes if we can change from one state to the other by swapping balls. You will... |
1,914,179 | <p>We have three balls in every move we can swap two of the balls.Prove that after an odd number of moves the permution of the balls are not same as they was first.</p>
<p><strong>My attempt</strong>:I cant do anything special but I find out that at last one of the balls is in its first place but the two other's are s... | Stefan4024 | 67,746 | <p>Consider the 3-cycle $(1,2,3)$ of the the symmetric group $S_3$. Note that swapping two balls, i.e. swapping the two numbers is equivalent to taking taking the inverse of the 3-cycle. In other words the action of swap can be defined as a function $f:S_3 \to S_3$ given by $f(a) = a^{-1}; \forall a \in S_3$.</p>
<p>U... |
4,631,224 | <p>This is the problem and the solution to it:</p>
<blockquote>
<p><span class="math-container">$$\begin{split}
\int \frac{1}{x-\sqrt{x}}dx& \\
u=\sqrt{x}-1&\quad du=\frac{1}{2\sqrt{x}}dx\\
\int \frac{1}{x-\sqrt{x}}dx
&=2\int\frac1u du\\
&=2\ln|u|+C\\
&=2\ln|\sqrt{x}-1|+C\\
\end{split}$$</span></p>
... | heropup | 118,193 | <p>The choice that was made is not the only possible choice, although it is perhaps the more "efficient" one.</p>
<p>A more obvious choice would be to select <span class="math-container">$$u = \sqrt{x}, \quad du = \frac{1}{2\sqrt{x}} \, dx.$$</span> Equivalently, we may write this as <span class="math-contai... |
4,631,224 | <p>This is the problem and the solution to it:</p>
<blockquote>
<p><span class="math-container">$$\begin{split}
\int \frac{1}{x-\sqrt{x}}dx& \\
u=\sqrt{x}-1&\quad du=\frac{1}{2\sqrt{x}}dx\\
\int \frac{1}{x-\sqrt{x}}dx
&=2\int\frac1u du\\
&=2\ln|u|+C\\
&=2\ln|\sqrt{x}-1|+C\\
\end{split}$$</span></p>
... | Lai | 732,917 | <p>There are options for substitution such as <span class="math-container">$x=\tan \theta$</span>.
<span class="math-container">$$
\begin{aligned}
I & =\int \frac{1}{\tan ^2 \theta-\tan \theta} \cdot 2 \tan \theta \sec ^2 \theta d \theta \\
& =2 \int \frac{1}{\tan \theta-1} d(\tan \theta) \\
& =2 \ln |\tan ... |
776,739 | <p>prove the inequality if you can: $\frac{1}{2}\cdot\frac{2}{3}\cdots\frac{2n-1}{2n}<\frac{1}{\sqrt{2n+1}}$</p>
<p>Thanks.</p>
| Madrit Zhaku | 34,867 | <p>$$\frac{1}{2}<\frac{2}{3},$$
$$\frac{3}{4}<\frac{4}{5},$$
$$.........................$$
$$\frac{2n-1}{2n}<\frac{2n}{2n+1},$$
$$\frac{1}{2}\cdot\frac{3}{4}\cdots\frac{2n-1}{2n}<\frac{2}{3}\cdot\frac{4}{5}\cdots\frac{2n}{2n+1}/\cdot \left(\frac{1}{2}\cdot\frac{3}{4}\cdots\frac{2n-1}{2n}\right)$$
$$\left(\f... |
942,225 | <p>I'm getting hung up on a probability question:
A car has six seats including the driver’s, which must be occupied by a driver. In how many ways is it possible to seat 4 people if all 4 can drive.</p>
<p>for one possible way you can choose the driver first and then choose seat options per person left so that would l... | André Nicolas | 6,312 | <p>Your first way of counting is perfectly good. </p>
<p>If you want to count another way, let us invent $2$ identical ghosts. The seats for them can be chosen in $\binom{5}{2}$ ways, since ghosts aren't allowed to drive. (There is a problem with taking their picture for the licence.) The rest of the seats can be fil... |
942,225 | <p>I'm getting hung up on a probability question:
A car has six seats including the driver’s, which must be occupied by a driver. In how many ways is it possible to seat 4 people if all 4 can drive.</p>
<p>for one possible way you can choose the driver first and then choose seat options per person left so that would l... | Walt | 652,109 | <p>There are three parts to this. There's always a driver and all 4 can drive so whatever the combination in the other 5 seats, you are going to multiply that combination by 4 to represent the 4 different sets of people who could be in the other seats.</p>
<p>Next, you have three people to fill five seats. You will ... |
1,441,932 | <p>I think I am making this problem far harder than it needs to be. Here is the statement: for each non-negative integer $n$, let $P_n$ be the space of real-valued polynomials of degree less than or equal to $n$. Find a Jordan Canonical basis for the map $T(f) = f' + f$.</p>
<p>My attempt: I let $\beta = \{1,x,x^2,...... | Martín-Blas Pérez Pinilla | 98,199 | <p>Maybe "controversy" isn't the word, but like the Gallic village, the <a href="https://en.wikipedia.org/wiki/Constructivism_(mathematics)" rel="noreferrer">constructivism</a> continues alive and defying the mainstream mathematics.</p>
<p>The entry <a href="https://plato.stanford.edu/entries/mathematics-constructive/... |
1,425,618 | <p>[Note that this is a reference request; I already know a couple of routine ways to prove the identity.]</p>
<p>In April I posted <a href="https://math.stackexchange.com/questions/1219016/indefinite-integral-with-sin-and-cos/1219317#1219317">this answer</a>. Then yesterday I had occasion to conjecture that in gener... | David Quinn | 187,299 | <p>We have $$a+b\tan(x-\phi)=a+\frac{b\tan x-b\tan\phi}{1+\tan x\tan\phi}$$
$$=\frac{\tan x(b-a\tan \phi)+(a-b\tan \phi)}{1+\tan x\tan \phi}$$</p>
<p>We can equate this to $$\frac{\frac ps\tan x+\frac qs}{\frac rs\tan x+1}$$ if we allow $$\frac ps=b-a\tan \phi$$ $$\frac qs=a-b\tan \phi$$ and $$\frac rs=\tan \phi$$</p>... |
1,425,618 | <p>[Note that this is a reference request; I already know a couple of routine ways to prove the identity.]</p>
<p>In April I posted <a href="https://math.stackexchange.com/questions/1219016/indefinite-integral-with-sin-and-cos/1219317#1219317">this answer</a>. Then yesterday I had occasion to conjecture that in gener... | Bangkockney | 264,240 | <p>You will probably find it in here:</p>
<p>Abramowitz, Milton; Stegun, Irene A., eds. (1972). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables.</p>
|
808,507 | <p>Let $a,b \in S_n$ and $ab=ba$ and $b$ moves some points that not moved by $a$. Is it true that $a$ and $b$ should be disjoint permutations? </p>
<p>EDIT:
We can consider $b=a^kc$ where $a$ and $c$ are disjoint, to construct a counterexample class. Do you know other constructions? </p>
| Santosh Linkha | 2,199 | <p>Consier the set of equations $$Ax^2+Ay^2+Bx+Cy+D=0 \tag{1} $$
$$Ax_1^2+Ay_1^2+Bx_1+Cy_1+D=0 \tag{2} $$
$$Ax_2^2+Ay_2^2+Bx_2+Cy_2+D=0 \tag{3} $$
$$Ax_3^2+Ay_3^2+Bx_3+Cy_3+D=0 \tag{4} $$
this is a $4\times4$ linear system and it must satisfy,
$$\left( \begin{array}{ccc}
x^2+y^2 & x & y&1 \\
x_1^2+y_1^2 &... |
3,954,579 | <p>I have two different questions, but they are related.</p>
<p>The first question is, Let <span class="math-container">$G$</span> be a finite abelian group. show that if <span class="math-container">$G$</span> contains (atleast) <span class="math-container">$2^n-1$</span> distinct elements of order 2, then there must ... | ancient mathematician | 414,424 | <p>For the first question.</p>
<p>Note that in an abelian group <span class="math-container">$o(xy)|o(x)o(y)$</span>, so that in <span class="math-container">$G$</span> the elements whose order is a power of <span class="math-container">$2$</span> form a subgroup, and this subgroup has at least <span class="math-contai... |
949,857 | <p>I have general questions about the group of isometries of a metric space.
-When is the isometry group of a space a lie group?
-when the isometry group is a Lie group, is there a relation between the one parameter sub-group of the isometry group and the geodesics of the metric space?</p>
<p>I am relatively new to th... | Robin Goodfellow | 176,079 | <p>I'm sure there are several cases I could give to answer this, but the two cases I would consider to occur the most commonly are the following:</p>
<ul>
<li>Any finite metric space, since all finite groups are (compact) Lie groups</li>
<li>By a theorem of Myers and Steenrod, Riemannian manifolds have Lie groups as i... |
180,053 | <p>In $C[0,1]$ the set $\{f(x): f(0)\neq 0\}$ is dense? I know only that polynomials are dense in $C[0,1]$, could any one give me hint how to show this set is dense?thank you.</p>
| S4M | 12,122 | <p>Your set $A = \{f\in C[0,1],f(0)\neq 0\}$ is dense in $C[0,1]$: let's take $f\in C[0,1]$ such as $f(0)=0$. We can consider the series $(f_n)$, where $$f_n(x) = \begin{cases}f(x) &\text{ if } x\geq\frac{1}{n} \\ \frac{1}{n}+n\times x\times \left(f\left(\frac{1}{n}\right)-\frac{1}{n}\right)& \text{ otherwise }... |
176,569 | <p>How to find a set $A \subset \mathbb{N}$ such that any sum of at most three Elements $a_i \in A$ is different if at least one element in the sum is different.</p>
<p><strong>Example with $|A|=3$:</strong> Out of the set $A := \{1,7,11\}$ follow 19 sums 1,2,3,7,8,9,11,12,13,14,15,18,19,21,22,23,25,29,33 which are al... | Tony Huynh | 2,233 | <p>Note that there is a bit of a discrepancy between $B_3[1]$ sets and your notion. For $B_3[1]$ sets, one considers sums with <em>exactly</em> 3 elements, instead of at most 3 elements. However, I do not think this will make a big difference. Basically the main point is that we can do much better than GH from MO's ... |
176,569 | <p>How to find a set $A \subset \mathbb{N}$ such that any sum of at most three Elements $a_i \in A$ is different if at least one element in the sum is different.</p>
<p><strong>Example with $|A|=3$:</strong> Out of the set $A := \{1,7,11\}$ follow 19 sums 1,2,3,7,8,9,11,12,13,14,15,18,19,21,22,23,25,29,33 which are al... | Javier | 56,340 | <p>The probabilistic method shows the existence of an infinite $B_h[1]$ sequence of positive integers $A$ with $A(x)>x^{1/(2h-1)+o(1)}$. The proof is easy for $h=2$ but involved for $h\ge 3$. In particular it gives the exponent $1/5$ for $h=3$.</p>
<p>Ruzsa improved the exponent $1/3$ to $\sqrt 2-1$ for $h=2$ and
v... |
98,913 | <p>I'm wondering to detect all the errors (i.e. their positions) in a codeword $(c_0, c_1, \cdots, c_{n-1})\in Q$ where $Q$ is an alphabet set with size $q$, i.e., to know whether $c_i$ is faulty or not, without asking for the exact initial value. Is it possible to achieve higher code rate than the ECC which corrects ... | Jyrki Lahtonen | 15,503 | <p>A bit of terminology:</p>
<p>An error-detection-code usually means something else. An error-detection-code is expected to raise a flag if something is wrong, IOW if the received sequence is not a valid codeword, or yet IOW at least one of the symbols $c_i$ is incorrect. It is <strong>NOT</strong> about telling that... |
96,576 | <p>I will call two graphs <span class="math-container">$G$</span> and <span class="math-container">$H$</span>, <span class="math-container">$r$</span>-equidecomposable (in analogy with <a href="https://en.wikipedia.org/wiki/Hilbert%27s_third_problem" rel="nofollow noreferrer">Hilbert's third problem</a>) if they can be... | Joseph O'Rourke | 6,094 | <p>Not an answer (cool question!),
just an excuse to show a pair of 2-equidecomposable graphs, one planar, one not:
<br />
<img src="https://i.stack.imgur.com/z667f.jpg" alt="Petersen Graph"><br /></p>
|
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